To solve for the vertex of the parabola given by the equation [tex]\( y = a(x-h)^2 + k \)[/tex] with the provided points, follow these steps:
1. Identify Symmetry:
A parabola exhibits symmetry around its vertex. The points with the same y-value on either side of the vertex can help identify the x-coordinate of the vertex. Specifically, the midpoint of the x-values of the points equidistant from the vertex will give the x-coordinate of the vertex.
2. Given Points:
The points provided are:
[tex]\[
\begin{array}{cc}
(0, 27), & (52.5, 12), & (105, 7), & (157.6, 12), & (210, 27) \\
\end{array}
\][/tex]
3. Calculate the x-coordinate (h):
Look at the points with the same y-values on either side:
- From [tex]\((0, 27)\)[/tex] and [tex]\((210, 27)\)[/tex]
Compute the midpoint of the x-values:
[tex]\[
h = \frac{0 + 210}{2} = 105
\][/tex]
4. Identify the y-coordinate (k):
At [tex]\( x = 105 \)[/tex], the corresponding y-value is given as:
[tex]\[
k = 7
\][/tex]
Thus, the vertex of the parabolic curve is [tex]\((105, 7)\)[/tex].
[tex]\(\boxed{105}\)[/tex]
[tex]\(\boxed{7}\)[/tex]
